Real-rooted atlas

A145324 → basis A013609 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-1 - 3\,n - 4\,n x - 3\,n^{2} - 8\,n^{2} x - 4\,n^{2} x^{2} - n^{3} - 4\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,n + 4\,n x + 2\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17411
2468224152
332610509681922536
42556122202022812752192049676
522212142564348380392864189504230401118564
62129761727168564304093593048056384315993632256028481368
722393442203718490696864200526984253192384176934784586229765160960809411480

All 15 rows

Technical details
Node id
n-d7afc1bf684a27bb · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3696a58d23a4e64f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         7,          4;
  2:        46,         82,          24;
  3:       326,       1050,         968,         192;
  4:      2556,      12220,       20228,       12752,         1920;
  5:     22212,     142564,      348380,      392864,       189504,        23040;
  6:    212976,    1727168,     5643040,     9359304,      8056384,      3159936,       322560;
  7:   2239344,   22037184,    90696864,   200526984,    253192384,    176934784,     58622976,      5160960;
  8:  25659360,  297722304,  1483451472,  4122056736,   6922720144,   7082041216,   4175988480,   1199720448,    92897280;
  9: 318540960, 4265031744, 25011185712, 83988405536, 176957151984, 240374517696, 207180558080, 105847538688, 26874224640, 1857945600;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(4*n^3+8*n^2+4*n)*x+(4*n^3+4*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(4*n^2+4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (4*n^3 + 8*n^2 + 4*n)*x + (4*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (4*n^2 + 4*n)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (4*n^3 + 8*n^2 + 4*n)*x + (4*n^3 + 4*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (4*n^2 + 4*n)*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.